Biomedical Engineering Reference
In-Depth Information
PTH in the RANK-RANKL-OPG pathway model by considering the effects
of PTH on bone cells as a regulator of RANKL and OPG production. They
assumed that all receptor-ligand binding reactions can be written in the fol-
lowing form:
A + B
j
=
C
j
( 7.1 5 )
where
A
and
B
j
are two reactants and
C
j
represents the molecule complex
formed. The index
j
was introduced in Pivonka et al. for cases where mole-
cule
A
is involved in several (say,
N
) reactions such as RANKL, both of which
bind to OPG and RANK (
N
= 2 in this case).
Using the principle of mass action kinetics, Pivonka et al. [3] presented the
concentration balance equation for all molecules as follows:
N
N
dA
dt
∑∑
=−
kAB
⋅ + +
k
CS
( 7.16 )
jf
,
j
j r
,
j
A
j
=
1
j
=
1
dB
dt
j
=−
kABkCS
⋅ + +
( 7.1 7 )
jf
,
j
j rj
,
B
j
dC
dt
j
( 7.18)
=
kABkC
⋅
−
jf
,
j
j rj
,
where
k
j,f
and
k
j,r
are the forward and reverse reaction rate constants, and
S
A
and
S
Bj
are source and sink terms, which express whether the overall contri-
bution adds or removes mass during a chemical reaction. They are the sum
of a production rate term (
P
A
,
P
Bj
) and a degradation term (
D
A
,
D
Bj
) [3]:
j
( 7.19)
SPD
=+
and
S
=+
PD
AA A
B
B
B
j
j
Then, Pivonka et al. assumed that binding reactions leading to upregu-
lation and downregulation of molecules are much faster than the cell
responses. As a result, the binding reactions can be assumed as quasisteady
states. Using the steady-state assumption, Equations (7.16)-(7.18) yield
S
=
0 nd
S
=
0
( 7. 2 0 )
A
B
j
To calculate
A,
the production rate of
A
is decomposed into an endogenous
term and an external “dosing term” and degradation of
A
is assumed to be
proportional to its concentration:
P
A
=
P
A,e
(
t
) +
P
A,d
(
t
)
( 7. 2 1)
=
DDA
A
( 7. 2 2)
A
